Cannot cut area out with AddModifier Boolean
When you delete a face you are making the cube physically impossible. It can't exist in real life. The reason for this is that it's sides are infinitely thin. With another word, it is not manifold anymore.

If you want to make it manifold, but open, add the solidify modifier before attacking it with boolean. Now the sides have a thickness again.



Now it is time for boolean. If you want, you can apply the solidify modifier first.